Coincidence detectors in Bell tests: How close is close enough? When is a coincidence a coincidence? We know that to identify entangled photons, the electronics is set to look for simultaneous clicks at opposite detectors. The size of the window is to some degree arbitrary. What I am wondering is:
If you run a Bell-type entanglement experiment where you get, for example, 100 coincidences per second with a 100 picosecond interval, what happens if you double the window to 200 picoseconds? Do you expect, in general, to get 200 coincidences per second? I can think of three possible answers and I wonder which one is most true:


*

*If your window is small enough, doubling the window doesn't matter because you are accurately identifying all available entangled particles. You still get 100 coincidences per second (or possible a few more for truly sporadic coincidence counts.)

*Double the window, double the coincidences. You get 200 per second.

*Double the window and you get four times the coincidences: 100 entirely in the first half of the window, 100 in the seconde half, and 200 "cross-coincidences". 400 per second in total.
I hope this question is meaningful and I wonder what the answer is.
 A: About $\pm 1$ nanosecond. The data can be visualized by plotting the difference between times of events at the two ends of the Bell experiment. The data from Gregor Weihs's experiment, Phys.Rev.Lett. 81, 5031 (1998), results in the attached graph (from an embedded postscript file, a much clearer PDF version is here), which shows a very distinct drop off in coincidence outside a small range of time differences.
The graph shows the time difference from each of 'Alice's events to the nearest 'Bob's event, with only events within $\pm 3$ nanoseconds shown. In the ten second time-slot shown, that's 14841 of a total of 388660 Alice events. In a very few cases, there are more than one Bob events within 3 nanoseconds of the marked Alice event. As you see, increasing the width of the coincidence window above $\pm 1$ nanosecond will have relatively little impact on the numbers of coincidences. The different graphic symbols indicate Alice's experimental result for the particular Alice event without showing Bob's experimental result.
I regret that I don't have comparable data for a different experiment, so I can't tell you whether different experimental apparatuses show different dependencies on the coincidence window width.
EDIT: As a response to the comments below, a different graph, done long enough ago that I'm not sure that it's from the same data, but I'm pretty certain that the features are generic. These graphs (the 2nd and 3rd graphs are just the first graph enlarged at the origin) plot the $i$'th sorted time difference between Alice's TOEs (Times Of Events) and the closest Bob's TOE for 8192 Alice events. The sorting of CTDs  (Closest TOE differences) makes the graph monotonely increasing. In this case, there are just 8192 Alice events instead of the 388660 Alice events in the full "longdist34 " dataset. Looking at the second graph, there are just less than 400 Alice events for which the corresponding CTD is less than about 0.1 microseconds; in the 3rd graph, there are 135 events with CTDs less than 1 nanosecond (the jumps are because at that time I was working with CTDs with 1 nanosecond resolution, whereas I later discovered that there is more resolution in the data; the resolution and systematic variations of the two clocks' time stamps at the times of events (which are what are recorded in the Alice and Bob datasets) are definitely issues to understand, however I ignore them here).
In the first graph, one can see an almost linear rise from 350 events until, say, about 5000 events, then there are relatively fewer events that have longer time differences. The cutoff is inevitable, because Bob's events are rarely separated by long time intervals, so that the largest CTD is relatively unlikely to be longer than whatever the average time separation is between Bob's events. In this case, the longest CTD for 8192 Alice events is about 0.14 milliseconds. I think there's an answer in these graphs to your supplementary questions in your Comments below (albeit a graphical explanation, not what I'd call a mathematical explanation).
A: Every coincidence test compares to the accidental chance rate. To find chance you use only one detector at a time to read the click rates, $R_1$ and $R_2$, for each detector. There is a time window, $\Delta t$, within which you will use to determine both the chance rate and the $\Delta t$ within which you will count as a coincidence in your experiment. The chance rate is $R_1 R_2 \Delta t$. When you run the experiment you use a time difference graph that should look like a peak in the middle of the graph, which is at zero click-time difference between the detectors. Here you will choose $\Delta t$ so that the bulk of that peak is used in your counting coincident clicks. Any peak in that graph means the experiment is reading better than chance. I have done many such experiments but they were with one "particle" emitted at a time instead of two that are used in a typical Bell test.
